Amazon cut prices on select items at Whole Foods on Aug. 28. On the first day, customer traffic spiked 31 percent from a year earlier. Traffic jumped 17 percent during the week after the price reductions and remained up 4 percent for the week ended Sept. 16, according to Thasos Group, which analyzes location data from millions of mobile phones.

Ten percent of Trader Joe's regular customers visited Whole Foods between Aug. 28 and Sept. 16, more than any other competing chain, Thasos Group said. Sprouts was No. 2 at 8 percent, followed by Target Corp at 3 percent and Costco Wholesale Corp and Safeway at 2 percent each.

Wal-Mart Stores Inc and Kroger Co, the leading U.S. grocery sellers with millions of shoppers, were the top sources of new customers after the price cuts at Whole Foods. They accounted for 24 percent and 16 percent of Whole Foods new customers, respectively, from Aug. 28 through Sept. 16. During that period, 15 percent of shoppers came from Costco, 11 percent were from Target and 5 percent were from Wal-Mart's Sam's Club.

The price reductions did not attract customers outside Whole Foods' traditional upper-income demographic. They also did not convince consumers to drive longer distances to shop at Whole Foods, said Thasos Group Chief Executive Greg Skibiski, who added that the dataset used for the Whole Foods competitive analysis includes 10 percent of the U.S. population.

"Amazon's acquisition of Whole Foods has the potential to be a gamechanger in the grocery space," Skibiski said.

Orbital Insight, which monitors business at about 65 percent of Whole Foods locations, reported a 5 percent year-over-year increase in car traffic to Whole Foods stores in the month since the acquisition closed on Aug. 28.

Foursquare, which analyzed the mobile phone movements of more than 2.5 million Americans, said traffic to Whole Foods was up about 13 percent the first week after the price cuts and remained up 8 percent after the second week.